This is a good question, which apparently has more than one answer.   This is also an interesting question.

A good place to look for an answer to this question is in

S. Bongers, Geometric quantization of symplectic and Poisson manifolds, M.Sc. thesis, Utrecht University, 2014:

http://dspace.library.uu.nl/bitstream/handle/1874/290019/Thesis.pdf

See Section 1.1, starting on page 47, where a simply connected Lie group G and a 3-dimensional smooth manifold for a basis for a classifying space BG such that gauge equivalence classes of principal G-bundles over the the BG space are in natural bijective correspondence with the set of homotopy classes of maps from the 3-space manifold to the BG space.   The smooth structure makes it possible to define the differential on the action functional and its critical locus, which is characterized by the Euler-Lagrange equations of motion.    A relation between the principal circle bundles BU(1)conn with connection, flat connections between on principal G-bundles of $\Sigma_2$ and $\Omega_{cl}^2$, the smooth stack of closed 2-forms (pp. 49-50, especially page 50).

Various forms of prequantum bundles are considered on page 51.   See, also, the pullback diagram on page 60.    Things are nicely put together in Remark 2.3.4, starting on page 61.  Higher symplectic geometry and symbolical manifolds are considered in great detail in Section 3, starting on page 72, especially Section 3.1, starting on page 73.   The is also an excellent Bibliography, starting on page 113.   See, for example,

D. Fiorenza, C.L. Rogers, U. Schreiber, Higher geometric prequantum theory, arXiv: 1304, 2013, no. 0237.

There are quite a few arXiv papers cited, each of which is downloadable from

http://arxiv.org/

especially see the arXiv page for symplectic geometry at

http://arxiv.org/list/math.SG/recent

thanks.